Quantum Optics of Soliton Microcombs

,1 ,1 ,1 1,2 ,1 Melissa A. Guidry∗ , Daniil M. Lukin∗ , Ki Youl Yang∗ , Rahul Trivedi and Jelena Vuˇckovi´c† 1E. L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, USA. 2Max-Planck-Institute for Quantum Optics, Hans-Kopfermann-Str. 1, 85748 Garching, Germany

Soliton microcombs — phase-locked microcavity frequency combs — have become the foundation of several classical technologies in integrated photonics, including spectroscopy, LiDAR, and optical computing. Despite the predicted multimode entanglement across the comb, experimental study of the quantum optics of the soliton microcomb has been elusive. In this work, we use second- order photon correlations to study the underlying quantum processes of soliton microcombs in an integrated silicon carbide microresonator. We show that a stable temporal lattice of solitons can isolate a multimode below-threshold Gaussian state from any admixture of coherent light, and predict that all-to-all entanglement can be realized for the state. Our work opens a pathway toward a soliton-based multimode quantum resource.

Kerr optical frequency combs1–3 are multimode states of light generated via a third-order optical nonlinearity in ˆ g0 an optical resonator. When a Kerr resonator is pumped Hint = δFWM(AmAnaˆ†aˆ† +A∗Anaˆ†aˆm +h.c.) − 2 j k k j weakly, spontaneous parametric processes populate res- m,n,j,kX onator modes in pairs. In this regime, Kerr combs can be (1) a quantum resource for the generation of heralded single where δFWM = δ(j+k m n) is the four-wave mixing 4–6 − − photons and energy-time entangled pairs , multiphoton (FWM) mode number matching condition and g0 is entangled states7–9, and squeezed vacuum10–12. When the nonlinear coupling coefficient. This approximation pumped more strongly, the parametric gain can exceed has successfully modelled the quantum correlations de- the resonator loss and give rise to optical parametric os- veloped by a coherent soliton pulse after propagating cillation (OPO) and bright comb formation. The modes through a fiber27,28, where the Kerr interaction is weak. of a Kerr comb can become phase-locked to form a stable, The DKS state, however, is itself generated by strong low-noise dissipative Kerr soliton (DKS). This regime of Kerr interactions where linearization can break down and Kerr comb operation has become the foundation of mul- yield unphysical predictions29,30. Experimentally verify- tiple technologies, including comb-based spectroscopy13, ing the validity of this model for the DKS is a prerequisite LiDAR14, optical frequency synthesizers15, and optical for exploring its application in quantum technologies. processors16. In this work, we use second-order photon correlation measurements to experimentally validate the linearized While the soliton microcomb has been mostly stud- model (Eq. 1) for describing the spontaneous parametric ied classically, it is nonetheless fundamentally governed processes in DKS states. We study the quantum for- by the dynamics of quantized parametric processes: each mation dynamics of Kerr frequency combs and identify resonator mode is coupled to every other mode through a class of DKS states — perfect soliton crystals31,32 — a four-photon interaction. The multimode coupled pro- which naturally isolate a subset of quantum optical fields cesses of the DKS resemble those of the well-studied 17,18 from the coherent mean-field (Fig. 1). We measure the synchronously pumped OPO . If the quantum pro- correlation matrix of the below-threshold modes to nu- cesses can be harnessed, soliton microcombs may open merically infer the multi-mode entanglement structure of a pathway toward the experimental realization of a 18–20 the state. We find that the DKS state may be engineered multimode quantum resource in a scalable, chip- as an on-chip source of all-to-all entanglement. integrated platform21,22. However, the quantum-optical properties of soliton microcombs have not yet been di- The observation of the full optical spectrum of a rectly observed: the first glimpse into non-classicality was Kerr frequency comb, including the above- and below- the recent measurement of the quantum-limited timing threshold processes, requires single-photon sensitivity arXiv:2103.10517v1 [physics.optics] 18 Mar 2021 jitter of the DKS state23,24. and a high dynamic range. For this purpose, we designed a single-photon optical spectrum analyzer (SPOSA) Linearization25 — the formal separation of the opti- using multipass grating monochromators and super- cal state into the mean field solution and the quantum conducting nanowire single-photon detectors (SNSPDs) fluctuations — is one approach to simplify the otherwise (PhotonSpot, Inc). The SPOSA has a broadband intractable many-body quartic Hamiltonian of the DKS. (> 200 nm) quantum efficiency of 20%, and close-in The result of this approximation is a quadratic Hamilto- dynamic range of > 140 dB at 0∼.8 nm. We generate ± nian, where the classical amplitudes Am, obtained for ex- Kerr frequency combs using microring resonators with ample through the spatiotemporal Lugiato-Lefever equa- 350 GHz free spectral range (FSR) fabricated in 4H- tion (LLE)26, drive the spontaneous parametric processes silicon carbide-on-insulator, a platform with favorable 33 34 between quantum modesa ˆj: nonlinear and quantum optical properties. With in- 2

FIG. 1: Linearized model for quantum optical ﬁelds in a DKS state A schematic depiction of a Kerr microresonator with a circulating perfect soliton crystal state. The full optical state is modeled as a coherent classical comb (blue) that drives the quantum comb (red) via spontaneous parametric processes.

FIG. 2: Single-photon spectroscopy of optical microcombs (a) Stages of frequency comb formation observed on the single-photon optical spectrum analyzer (SPOSA). Dashed line indicates noise floor of a commercial optical spectrum analyzer (-80 dBm). (b) The single soliton state. (c) A 7-FSR soliton crystal state, observed in a different device. trinsic quality factors (Q) as high as 5.6 106, low thresh- 3 modes with very low photon number aj†aj < 10− . old OPO and low-power soliton operation× (0.5 mW and The quantum fluctuations generated inD theE DKS state 2.3 mW, respectively) are achieved. We use the SPOSA and responsible for its quantum-limited timing jitter23,24 to characterize the Kerr microcomb spectra through all 2 are obscured in the single-soliton spectrum. A perfect stages of DKS formation (Fig. 2a,b). The sech soliton soliton crystal31,32 (henceforth referred to as a soliton envelope is seen to persist at the tails of the DKS, in crystal), however, reveals these quantum fluctuations 3

FIG. 3: Quantum coherence of parametric oscillation (a) The near-threshold spectrum reproduced from (2) Fig. 2a. Vertical dashed lines indicate the modes where the primary comb will form. (b) Measured gauto(τ) on different modes shows the dispersion-dependent parametric gain variation throughout the comb. (c) Observation of asymptotic growth of coherence near the OPO threshold. The eﬀective parametric gain λ, extracted from a numerical ﬁt to Eq. 2 is plotted against the detected count rate on mode µ = +5. (d) At the highest photon count rate, (2) gauto(τ) reveals coherence-broadening which corresponds to threshold proximity of 0.99896(5).

(Fig. 2c). cess (analogous to lasing). In a Kerr frequency comb close The formation of DKS from a below-threshold quan- to but below the OPO threshold (Fig. 3a), both regimes (2) tum frequency comb begins with the transition of a spon- can be observed simultaneously. We measure gauto(τ) taneous FWM process into a stimulated FWM process using two SNSPD detectors in a Hanbury Brown and at the onset of OPO. This transition can be observed Twiss configuration, and observe the dispersion-induced through the second-order correlation function, g(2)(τ). variation in λ for different signal-idler pairs (Fig. 3b).36. Using input-output theory35 we derive the exact form of Mode µ = +9, far away from the pump, displays oscil- (2) (2) g (τ) for a general two-mode parametric process. The lations in gauto(τ), signifying poor phase-matching. In general derivation is presented in the Supplementary In- contrast, mode µ = +5 shows a substantial coherence information. For signal and idler modes of equal linewidths crease, which correctly predicts that it will seed the for- κ, the auto-correlation is given by mation of the primary comb. To observe the asymptotic coherence growth at threshold, we repeatedly sweep the κτ 2 (2) e− κ pump laser detuning through the OPO threshold con- gauto(τ) = 1 + sinh(λτ) + λ cosh(λτ) (2) λ2 2 dition, while synchronously acquiring the photon count h i rates and two-photon correlations. Through the numer- where λ = g2 δ2 is the effective parametric gain, 2 − ical fit to Eq. 2, λ is extracted and plotted against the g = g0 A0 is the mode coupling strength, and δ is the | | p 2 2 measured count rate (Fig. 3c). The maximum recorded detuning. Here, two regimes are notable: when δ > g , 35 (2) coherence broadening exceeds the cavity coherence by λ is imaginary which gives rise to oscillations in gauto(τ), nearly three orders of magnitude (Fig. 3d), which indi- corresponding to the double-peaked photon spectrum for cates that the state is approaching the critical point at 25,30 a strongly detuned parametric process ; when λ ap- which the linearized model would break down.29,30 (2) proaches κ/2, the gauto(τ) coherence (decay time) in- creases asymptotically, reflecting the transition of the In the linearized model of an above-threshold Kerr spontaneous FWM process into a stimulated FWM pro- comb, a resonator mode may be occupied by both a co- 4

FIG. 4: Formation dynamics of secondary combs (a) A SPOSA spectrum of a secondary comb state. (b) A graphical representation of the stages of secondary comb formation. (c) The auto-correlation is measured at mode µ = +4 for each comb state. In state 1, only non-degenerate pair generation contributes to mode µ = +4. In state 2, simultaneous degenerate and non-degenerate spontaneous pair generation is present. In state 3a, subcombs begin to merge and two-photon correlations reveal the interference of quantum fluctuations with the coherent state. The data are overlaid with the fit to input-output theory. In state 3b, the coherent state dominates and only the RF beat note is observed. herent state and quantum fluctuations. In theory, the state 3a). The Fourier transform of the auto-correlation interference of the coherent state and the quantum fluctu- shows three peaks, at 0, ∆/2, and ∆; they represent, re- ations can be revealed through g(2)(τ), but the intensity spectively, the two-photon bunching of spontaneous pair of the former is usually orders of magnitude greater (as generation, the interference of the quantum fluctuations can be seen in the spectra of the primary comb and the with the bichromatic coherent state, and the coherent RF soliton crystal, Fig. 2), making the experimental obser- beat note36,37 of the coherent state. The details of the vation difficult. An exception can be found in the forma- theory used to model the interference are presented in the tion of secondary combs. We characterize the quantum Supplementary Information. As the subcombs continue formation dynamics of secondary combs through second- to merge, the coherent light drowns out the spontaneous order photon correlations (Fig. 4), complementing ear- parametric processes, and the photon correlations corre- lier classical studies36,37. By monitoring mode µ = +4, spond to the interference of two weak coherent sources.38 which is equidistant from the pump and the primary side- In a DKS state, the coherent comb is phase-locked and band, we observe simultaneously the degenerate and non- time-independent in the group-velocity reference frame, degenerate spontaneous pair generation processes, as well yielding a time-independent Hamiltonian for the quan- as the merging of the coherent subcombs. Notably, at the tum optical fields (Eq. 1). In contrast, a comb that onset of subcombs merging, the intensities of the (bichro- is not phase-locked will produce a Hamiltonian with a matic) coherent state and the quantum optical fields be- time-dependent drive. To observe this effect in exper- come comparable, and the signature of their interference iment, we consider a microring resonator which sup- in two-photon correlations is readily observed (Fig. 4c, ports three distinct states with a 2-FSR spacing: one 5

FIG. 5: Quantum correlations in non-phase-locked combs and perfect soliton crystals (a)A SPOSA spectrum of a 2-FSR soliton crystal. (b) Left: The measured g(2)(τ) at mode µ = 15 for different 2-FSR states observed in the device (inset shows an OSA spectrum of the measured state). Right:− The RF beat note measured on a photodetector at mode µ = 6 (red), and the Fourier transform of the measured g(2)(τ) (blue). Bottom panel includes the photodetector noise floor,− corroborating the low-noise soliton state. (c) The second-order correlations matrix for the below-threshold modes in the 2-FSR soliton crystal. Left: theoretical model; Right: experimental data. (d) The logarithmic negativity (EN ) matrix calculated for the 2-FSR soliton crystal assuming 10 increased out-coupling of the below-threshold modes. No pairwise entanglement is predicted in the device with unmodified× waveguide coupling.

(2) (2) state is a 2-FSR soliton crystal (Fig. 5a); the others are g (τ) = g (t, τ) t on mode µ = 15 while simultane- non-natively spaced secondary combs36 in the process of ously measuring the RF spectrum of− mode µ = 6 on a merging. These secondary combs are non-phase-locked photodetector. Whereas the soliton crystal two-photon− states, which manifests in the frequency domain as poly- correlations are time-independent far from zero time de- chromatic comb teeth. For each 2-FSR state, we measure lay, the correlations of a non-soliton state exhibit oscilla- 6 tions whose Fourier transform matches the RF spectrum In conclusion, we have investigated the quantum for- measured on the photodetector. Such temporal dynamics mation dynamics of DKS states and their quantum corre- may be modelled via Floquet theory39. We note that sim- lations. The experimental methods introduced here can ilar temporal oscillations in g(2)(τ) have been observed serve as a starting point for exploring higher-order cor- in Floquet-driven two-level systems40. rections to the linearized model both in the DKS and The soliton crystal state offers an excellent opportu- at near-threshold critical points. Applications of DKS nity to experimentally verify the linearized model for the states in quantum technologies will be informed by fur- DKS state. The mean-field solution (complex amplitudes ther theoretical and experimental characterization of the 17 Am in Eq. 1) of the soliton crystal can be readily com- intra-comb entanglement. Once target supermodes and puted via the LLE; the below-threshold modes comprise nullifiers18,44 are identified for the system, dispersion the quantum fluctuations driven by the mean-field soliton engineering45,46 can be employed to design the desired without any admixture of coherent light and, crucially, system Hamiltonian. The recent demonstrations of soli- they are decoupled from the the quantum fluctuations of ton generation in CMOS-foundry photonics47,48, efficient the above-threshold modes by the mode matching con- on-chip frequency translation49, integrated detection of 50 51 dition δFWM. We measure the correlations between all squeezed light , and on-chip squeezed microcombs lay pairs of below-threshold modes of the 2-FSR soliton crys- out a clear path toward developing the DKS state as tal, and compute the theoretically predicted second-order a resource for continuous variable quantum information correlations matrix for the measured device parameters processing52–55. (Fig. 5c). We note that the only free parameter in the The large-scale multi-mode entanglement possible in a model is the pump laser detuning (within the soliton lock- DKS state may also find applications in discrete variable ing range). The agreement of the model with the ex- quantum computation protocols operating under con- periment suggests that the quadratic Hamiltonian of the tinuous wave (CW)5,9 or pulsed7,8 drive. Although the linearized model is indeed appropriate for describing the soliton is traditionally operated in the CW regime, the photon statistics of the quantum optical field generated recent study of solitons driven by a pulsed pump may in the below-threshold modes of a DKS state. enable the implementation of time-bin entanglement We perform an analysis of the multipartite entangle- protocols56 and the study of the temporal dynamics of ment across the 2-FSR soliton crystal in the resonator quantum correlations57 under Floquet drive. Pulsed mode basis by computing the logarithmic negativity41, operation can also be achieved with a CW-driven soliton EN , for all mode pairs. We find that in this basis, no via a time-dependent coupling constant g0(t) (Eq. 1), via pair-wise entanglement is predicted in the measured de- the rapidly-tunable photonic molecule configuration42, vice. We then consider a modified device architecture, to realize multimode control of spectral and temporal where the below-threshold modes are overcoupled to the entanglement in a single integrated photonic device. output waveguide via a photonic molecule configuration Finally, we note that with the recent demonstration (two coupled microresonators)42,43. This architecture is of Pockels soliton microcombs58, our results may be advantageous because it allows for the efficient extrac- further extended to interactions between second- and tion of the quantum optical fields from the device while third-order parametric processes. simultaneously filtering them from the coherent fields, all without impacting the soliton crystal (see Supplemen- ∗These authors contributed equally. tary Information). For this system, we numerically ob- † [email protected] serve all-to-all entanglement along the signal-idler diag- onal of the pump (Fig. 5d). Such entanglement struc- Acknowledgments We gratefully acknowledge discus- ture is consistent with the all-to-all connectivity in the sions with John Bowers, Tian Zhong, Lin Chang, 2-FSR soliton crystal Hamiltonian. In (N > 3)-FSR soli- Chengying Bao, Boqiang Shen and Avik Dutt. This work ton crystals, the below threshold modes are not all-to-all is funded by the Defense Advanced Research Projects coupled. Instead, the modes are divided into disjoint Agency under the PIPES program. M.A.G. acknowl- sets grouped by the value µ mod N , as per the mode- edges the Albion Hewlett Stanford Graduate Fellow- matching condition. Thus,| N/2 non-interacting| sub- ship (SGF) and the NSF Graduate Research Fellowship. sets (each internally all-to-allb coupled)c are expected in D.M.L. acknowledges the Fong SGF and the National De- an N-FSR soliton crystal. Indeed, for the 7-FSR soliton fense Science and Engineering Graduate Fellowship. Part crystal state presented in Fig. 2d, we experimentally con- of this work was performed at the Stanford Nanofabrica- firm three disjoint all-to-all correlated sets of modes (see tion Facility (SNF) and the Stanford Nano Shared Facil- Supplementary Information). ities (SNSF).

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CONTENTS

I. Linearized model of the soliton microcomb 2 A. Linearization 2 B. Input-output formalism 2 C. Second-order photon correlations below threshold 3 D. Numerical analysis of OPO threshold 4 E. Second-order photon correlations in merging secondary combs 6 F. Characterizing entanglement between cavity modes in soliton crystals 6

II. Experimental setup 8

III. Silicon carbide soliton microcomb 9 A. Sub-mW parametric oscillation threshold 9 B. Demonstration of a Silicon Carbide soliton microcomb 9

IV. Soliton crystals 11 A. 2-FSR soliton crystal 11 B. Photonic molecule analysis 11 C. 7-FSR soliton crystal 13

References 13 arXiv:2103.10517v1 [physics.optics] 18 Mar 2021 2

I. LINEARIZED MODEL OF THE SOLITON MICROCOMB

In this section, we detail the linearized model and the calculation of second-order photon correlations. First, we begin with a general quartic Hamiltonian for a Kerr resonator with a single pump; we linearize the model to produce a quadratic Hamiltonian, where modes are coupled through the classical amplitudes which can be described by the LLES1. Next, we deﬁne the input-output equations for the open quantum system, which we use to calculate the second-order correlation functions in a below-threshold comb. We numerically model the experiment of Fig. 3 from the main text to demonstrate the interface of the LLE with the input-output formalism. We then extend our model to describe photon correlations in the presence of bichromatic coherent light. Finally, we describe how to characterize entanglement (the logarithmic negativity, EN ) between diﬀerent cavity modes starting from the Heisenberg equations.

A. Linearization

We consider the most general system Hamiltonian for four-wave mixing between cavity modes with coherent drive of the pump mode, µ = 0:

1 iω t iω t Hˆ = ω aˆ† aˆ g δ[µ + ν j k]â† aˆ† aˆ aˆ + α (â e p +a ˆ†e− p ) (1) sys µ µ µ − 2 0 − − µ ν j k 0 0 0 µ X µ,ν,j,kX where δ[µ + ν j k] is the Kronecker delta which enforces the four-wave mixing mode-matching condition. Here, ω − − µ is the resonance frequency of cavity mode µ and ωp is the frequency of the coherent pump driving the central mode. The nonlinear coupling coefficient

2 ~ωpcn2 g0 = 2 n0Veff represents the per photon frequency shift of the cavity due to the third-order nonlinearity of the cavity: ~ is the reduced Planck’s constant, n2 is the nonlinear refractive index, n0 is the material index, and Veff is the effective mode volume of the resonator. The amplitude of the drive field is

κcPwg α0 = s ~ωp where κc is the coupling rate of the cavity to the input waveguide and Pwg is the power in the input waveguide. To linearize the system, we write a formal separation of the optical state into the mean field solution (assumed to be a coherent state) and the quantum fluctuations:a ˆµ(t) αµ(t) +a ˆµ(t), where αµ(t) is the complex amplitude of the coherent state inside the cavity. Moving into the reference→ frame which removes explicit time dependence from the classical coupled mode equationsS2, we apply a unitary transformation using Uˆ(t) = eiRtˆ where Rˆ = µ(ωp + D1µ)âµ† aˆµ. We enter the rotating frame of an evenly-spaced frequency ruler, with spacing D1, centered at the central pump mode. Defining δµ = ωµ ωp D1µ, and keeping only quadratic terms, we arrive at a Hamiltonian whichP includes the interaction Hamiltonian− Eq.− (1) of the main text:

spontaneous pair generation g0 Hˆ = δ aˆ† aˆ δ[µ + ν j k](A A aˆ†aˆ† + A∗A aˆ†aˆ +h.c.) (2) sys µ µ µ − 2 − − µ ν j k k ν j µ µ µ,ν,j,k X X z }| XPM{ & Bragg scattering

| {z } S1 where Aµ are the complex-valued ﬁeld amplitudes, described using the Lugiato-Lefever equation . The last term describes cross-phase modulation (XPM) when µ = j, and four-wave mixing Bragg scattering otherwise.

B. Input-output formalism

The following calculations are in the Heisenberg picture. To describe the open system, we allow our cavity modes to couple to a bath. For a resonator with a large ﬁnesse, a Markovian approximation can be made on each mode 3 independently

ˆ ˆ Hbath = ωbµ† (ω)bµ(ω)dω (3) µ X Z with a bath-coupling Hamiltonian

κµ V = aˆ† (ω)ˆb (ω)dω (4) 2π µ µ µ r X Z where κµ is the total loss rate for cavity mode µ. Starting from the Hamiltonian H = Hsys + Hbath + V , we use the Heisenberg equations of motion aˆ˙ µ = i[ˆaµ, Hˆ ] to write down quantum coupled mode equations for this system which resemble the classical coupled mode equations:−

daˆµ(t) κµ = iδ + aˆ (t) + ig δ[µ + ν j k]A A aˆ† (t) dt − µ 2 µ 0 − − j k ν ν,j,kX + 2ig0 δ[µ + j ν k]A∗Akaˆν (t) i√κµˆbin,µ(t) (5) − − j − ν,j,kX Each bath-cavity mode pair has an associated input-output relation:

ˆbout,µ(t) = ˆbin,µ(t) i√κµaˆµ(t) (6) − We deﬁne the following 2n-dimensional vectors describing n quantum modes in the frequency domain:

ˆ ˆ aˆ1(ω) bin,1(ω) bout,1(ω) . . . . .  .   .   .  ˆ ˆ aˆn(ω) ¯  bin,n(ω)  ¯  bout,n(ω)  a¯(ω) =   bin(ω) = ˆ  bout(ω) = ˆ  (7) aˆ1†( ω) bin† ,1( ω) bout† ,1( ω)  −.   −   −   .   .   .   .   .   .        aˆn† ( ω) ˆb† ( ω) ˆb† ( ω)  −   in,n −   out,n −        We can define a matrix N(ω) from our coupled mode equations relating the output fluctuations to the input fluctuations:

¯bout(ω) = N(ω)¯bin(ω) (8)

C. Second-order photon correlations below threshold

We derive analytic expressions for two-photon correlations between quantum modes which interact with a single pumped modeS3. In this case, our Hamiltonian is simple, and we only consider interactions between pairs of modes ( µ, +µ) centered around the pump. For this analysis, we choose D1 = (ω+ ω )/2µ for which δ+ = δ = δ. − − − − ˆ Hsys = δ(ˆa† aˆ +a ˆ+† aˆ+) + ig(ˆa aˆ+ aˆ† aˆ+† ) (9) − − − − − where g = g A 2, which includes both the Kerr coupling strength and the intensity in the pumped cavity mode. 0| 0| Assuming κ+ = κ = κ, the Heisenberg equations read − daˆ (t) κ ± = iδ + aˆ (t) gaˆ† (t) i√κˆbin, (t) (10) dt − 2 ± − ∓ − ± 4 which gives us the following matrix N(ω):

κ(κ/2 i(δ+ω)) gκ 1 + (g2 δ2) −(κ/2 iω)2 0 0 (g2 δ2) (κ/2 iω)2 − − − κ(κ/2 i(δ+ω)) gκ − − −  0 1 + (g2 δ2) −(κ/2 iω)2 (g2 δ2) (κ/2 iω)2 0  N(ω) = − gκ− − −κ(κ/−2+i(δ− ω)) 0 1 + − 0  (g2 δ2) (κ/2 iω)2 (g2 δ2) (κ/2 iω)2   gκ − − − − − − κ(κ/2+i(δ ω))   (g2 δ2) (κ/2 iω)2 0 0 1 + (g2 δ2) (κ/−2 iω)2   − − − − − −    We calculate the two-photon correlation function:

G(2)(t + τ, t) g(2)(t + τ, t) = (11) ij ˆ ˆ ˆ ˆ bout,i† (t)bout,i(t) bout,j† (t + τ)bout,j(t + τ) D ED E where

(2) ˆ ˆ ˆ ˆ G (t + τ, t) = bout,i† (t)bout,j† (t + τ)bout,j(t + τ)bout,i(t) . (12) D E The expectation value is taken with respect to the initial vacuum state. The time-domain output bath operator is related to the frequency-domain input bath operator via:

n ˆ 1 ∞ iωtˆ 1 ∞ iωt ˆ ˆ bout,i(t) = dωe− bout,i(ω) = dωe− Nik(ω)bin,k(ω) + Ni(k+n)(ω)b† ( ω) (13) √2π √2π in,k − Z−∞ Z−∞ kX=1 (2) From here, gij (t + τ, t) may be calculated using the commutation relations of the frequency-domain input bath operators:

ˆb (ω)ˆb† ( ω0) = δ δ(ω + ω0) ˆb† ( ω)ˆb (ω0) = 0 in,i in,j − ij in,i − in,j D E D E After applying the commutation relations, one arrives at an expression where each term is a product of Fourier transforms. Deﬁning λ = g2 δ2, the result is: − p e κτ κ 2 g(2) (τ) = 1 + − sinh(λτ) + λ cosh(λτ) (14) ++ λ2 2 κτ h i 2 (2) e− κ δ κ g+ (τ) = 1 + λ i sinh(λτ) + iδ cosh(λτ) (15) − g2 − 2 λ 2 −

(2) (2) (2) (2) and g++(τ) = g (τ), g+ (τ) = g +( τ). −− − − −

D. Numerical analysis of OPO threshold

To compute the second-order correlation matrix of the quantum optical fields in the presence of the above-threshold Kerr comb, we combine the LLE simulation with input-output theoryS2. The basic test case for the self-consistency of the joint LLE and input-output modelling is the OPO threshold condition: specifically, when the laser is tuned from blue to red to model the experiment, the asymptotic bandwidth narrowing in the below-threshold mode (as computed via input-output theory) and the formation of primary combs (as computed via the LLE) should happen simultaneously. We numerically reproduce the near-threshold behavior presented in Fig. 3 of the main text. The result of the combined LLE and input-output theory simulation is presented in Fig. S1a. The modes µ = 5 indeed exit ± the regime of validity of the input-output formalism (when Qeff has positive real eigenvalues) at the onset of the LLE threshold. Although coherent comb light is present in other modes above threshold, only µ = 5 modes cannot be simulated with the presented linearization approach. A qualitative agreement between simulation± and experimentally measured correlations (shown in Fig. 3b of the main text) is seen at the simulated detuning of 9.3 MHz, Fig. S1c. − 5

FIG. S1: Numerical analysis of OPO threshold a Top: Dependence of the pump intensity in the microring (in units of photon number) on laser detuning. Detuning is given relative to the OPO threshold; a higher laser frequency corresponds to a more negative detuning with respect to the OPO condition. The LLE simulation (red) matches the analytic expression for the cavity mode in the presence of the Kerr nonlinear resonance shift (dashed black line) up to the threshold point, where the pump mode becomes depleted. Middle: Spectrum of the classical Kerr comb in the ring resonator, computed via the LLE, showing the formation of the 5-FSR primary comb. The scale represents log10(# of photons). Bottom: The photon spectrum of the below-threshold modes computed via input-output theory. For each mode, a spectral window of 2 GHz is shown. At threshold, modes µ = 5 exit the regime of validity of the linearized model and are excluded± from the input-output simulation. The scale± represents photon number spectral density, log10(# of photons per Mrad/second). b Evolution of the spectrum of three select modes near threshold (same scale as (b)). Above threshold, the spectra exhibit additional features generated by the (2) additional parametric processes driven by the primary comb lines. c Computed gauto(τ) at a detuning of 9.3 MHz (indicated as a dashed line in (b)). − 6

E. Second-order photon correlations in merging secondary combs

In this section we describe the modelling of the interference in g(2)(τ) of stimulated and spontaneous FWM presented in Fig. 4 of the main text. The operator ˆbout(t) is:

ˆb (t) = ˆb (t) i√κ[α(t) +a ˆ(t)], (16) out in − or, in the frequency domain,

ˆb (ω) = N(ω)ˆb (ω) i√κα(ω). (17) out in − Since the coherent ﬁeld in the mode is bichromatic,

iω1t iω2t α(t) = Acoh,1e− + Acoh,2e− . (18)

We note that in the presence of a bichromatic coherent state, there is a distinction between g(2)(τ) and the experimentally-measured correlations. Speciﬁcally, the measured correlations are the time-averaged two-photon coin- cidences G(2)(τ) = G(2)(t + τ, t) , normalized to the mean value at τ : t → ∞

∞ (2) (2) G (t + τ, t) dt gexp(τ) = −∞ (19) ∞ ∞ (2) limT R T G (t + τ, t) dt dτ →∞ −∞ (2) R R To model the gexp(τ) presented in Fig. 4 of the main text, we consider a system of three coherent drives (A 8,A0,A+8), − and four quantum modes (â 12, aˆ 4, aˆ4, aˆ12). The effect of other coherent driving modes is assumed negligible, because − − their amplitude is much smaller. The fit parameters in the model are: 1) two pump amplitudes (A0, and A+8 = A 8); − 2) two coherent state amplitudes (Acoh,1 and Acoh,2); 3) two coherent state frequencies (ω1 and ω2); and 4) the pump laser detuning δp. The fit is presented in Fig. 4, state 3a of the main text.

F. Characterizing entanglement between cavity modes in soliton crystals

Denoting the modes under consideration by aµ for µ 1, 2 ...N and assuming a quadratic Hamiltonian. In steady state (t ), the density matrix describing any two∈ modes { α and} β is described by a Gaussian Wigner function: → ∞

T 1 T Wα,β(qα,β = [xα, pα, xβ, pβ] ) = exp (qα,βΣα,βqα,β) (20) π Det[Σα,β]

T p where Σα,β = qα,βqα,β W is the 4 4 steady state correlation matrix formed from weyl ordered operators. The entanglement measureh betweeni these× two modes can be computed as a log-negativity of this matrix, deﬁned by

= max[0, log(√2η)] (21) Eα,β − where

η = Θ Θ2 4Det(Σ ) (22) − − α,β r q and

Θ = Det(Σ ) + Det(Σ ) 2Det(C) (23) α β − and Σα, Σβ and C are deﬁned as diﬀerent blocks of Σα,β

Σα C Σα,β = T (24) C Σβ 7

To compute the correlation matrix Σα,β from a quadratic Hamiltonian, it is convenient to express the correlation elements in terms of annihilation operators. We can immediately note that

1 x x = a a + a† a† + a† a + a a† (25a) h α βiW 2 h α βi h α βi h α βi h α βi 1 p p = a a + a† a† a† a a a† (25b) h α βiW −2 h α βi h α βi − h α βi − h α βi i x p = p x = δ + a† a† a a + a† a a a† (25c) h α βiW h β αiW 2 α,β h α βi − h α βi h β αi − h β αi The correlators for the annihilation operators required above can be easily calculated from the input-output formalism. Recall that for a quadratic, time-invariant Hamiltonian, the Heisenberg equations read

d a(t) a(t) bin(t) = Qeﬀ + M (26) dt a†(t) a†(t) b† (t) in

Physically, we expect eigenvalues of Qeff to all have negative real part so as to have a well defined steady state. We then obtain by integrating the above equations that as t → ∞ t t a(t) Q (t τ) bin(τ) Λ(t τ) 1 bin(τ) = e eff − M dτ = Xe − X− M dτ (27) a†(t) b† (τ) b† (τ) Z0 in Z0 in 1 where we can define the eigenvalue decomposition Qeff = XΛX− . It is now straightforward to calculate

t a(t) Λ(t τ) 1 Λ∗(t τ) lim a†(t) a(t) = lim Xe − X− MM †X−†e − X†dτ = XNX† (28) t h a†(t) i t →∞ →∞ Z0 where N is a matrix whose elements are given by

1 [X− MJM †X−†]i,j Ni,j = (29) λi + λj∗ deﬁned in terms of the eigenvalues of Q , λ , where J is a 2N 2N matrix: eﬀ i × I(N) 0 J = 0 0 8

II. EXPERIMENTAL SETUP

The experimental setup is presented in Fig. S2. The device is operated at 4 K to reduce the thermo-optic responseS4. In the design of the single-photon optical spectrum analyzer (SPOSA), we focused on broadband operation in order to image all parts of the frequency comb, which extends beyond the operation range of standard fiber-based filtering components. For this reason, a free-space monochromator approach was chosen. Blazed gratings (600 grooves/mm) optimized for 1.5 µm operated in the Littrow configuration have 75 85% efficiency across the range of operation (1300- 1700 nm). In a single-pass monochromator, dynamic range is limited− to 70 dB by roughness-induced scattering from the grating surface. By operating the monochromator in a two-pass configuration,∼ where the forward and return beams do not overlap on the grating, the dynamic range is doubled. To further increase the dynamic range, a second, single-pass monochromator can additionally be used. The double-pass monochromator configuration has dynamic range in excess of 180 dB and total peak efficiency as high as 55%. The superconducting nanowire single photon detectors (SNSPDs) are optimized for broadband operation with efficiency exceeding 80% from 1.0 1.6 µm. −

FIG. S2: Single-photon OSA Experimental setup. A silicon carbide microring resonator interfaced with inverse- designed vertical couplersS5 is mounted in a closed-cycle cryostat. Automated locking of a desired microcomb state is performed using active feedback by controlling the laser wavelength and power. A free-space tunable two- pass monochromator (MC) and double monochromator (DMC) serve as narrow band-pass ﬁlters with rejection of >130 dB and >180 dB, respectively. The dynamic range of the superconducting nanowire single photon detectors (SNSPDs) is extended from 60 dB to 180 dB via variable optical attenuators (VOA). A single SPOSA is used for spectroscopy and two SPOSAs are used for photon correlation measurements. Photon detection events are recorded with a timing module TimeTagger Ultra from Swabian Instruments. 9

III. SILICON CARBIDE SOLITON MICROCOMB

In this section, we describe the first demonstration of a soliton microcomb in a 4H-silicon carbide (SiC) microresonator. The CMOS-compatible fabrication process is described inS5,S6. SiC possesses a high linear and nonlinear S5 15 2 refractive indices (n = 2.6 and n2 = 6.9 10− cm /W at 1550 nm), which makes it suitable for highly efficient, compact nonlinear photonic devices. However,· the tight confinement and high material index of integrated waveg- uides make them susceptible to scattering losses caused by surface roughness. We demonstrate the fabrication of SiC microresonators with smooth sidewalls and strong confinement with record-high quality (Q) factors. The fabricated microring resonators have a radius of 100 µm, height of 500-600 nm, and width of 1850 nm.

A. Sub-mW parametric oscillation threshold

The eﬃciency of the Kerr nonlinear interaction improves with higher quality factors of the optical resonator. For example, the threshold relation for optical parametric oscillation in a microresonator can be expressed in the following form:

πnω0Aeff 1 Pth = 2 (30) 4η n2 D1Q where Q denotes the total Q factor (intrinsic loss and loading included) with pump mode frequency ω0, Aeff is the effective mode area, η is the cavity-waveguide coupling strength, and D1 is the free-spectral range (FSR) in units of rad/s. Zero detuning of the laser frequency with respect to the pump mode frequency is assumed. The parametric oscillation threshold is inversely proportional to the square of the Q factor. Fig. S3 shows a sub-milliwatt (approximately 510 µW) parametric oscillation threshold of a SiC optical resonator featuring an intrinsic Q factor of 5.6 million with a 350 GHz FSR.

FIG. S3: Sub-mW parametric oscillation threshold power (a) SiC parametric oscillation induced by pump- ing at the wavelength of 1553.3 nm. Top panel shows OPO just above the threshold power (510 µW total power in the waveguide). Middle and lower panels show measured optical spectra with loaded pump power of approximately 570 and 600 µW, respectively. (b) High-resolution scan of the fundamental TE mode with a loaded (intrinsic) quality factor of 3.19 (5.61) million. The mode is seen to be nearly critically-coupled to the waveguide. The scan laser wavelength is calibrated using a wavemeter, and the red curve is a ﬁt to a Fano lineshape. The asymmetry of the resonance shape is attributed to interference with back-reﬂection of the vertical couplers.

B. Demonstration of a Silicon Carbide soliton microcomb

Coherently pumped solitons in optical microresonators form as a result of the balance of the Kerr nonlinear shift and the cavity dispersion, as well as the parametric gain and the cavity loss. The soliton-forming mode family (in particular for bright solitons) in a microresonator must feature anomalous dispersion and minimal distortion of the dispersion 10

FIG. S4: SiC soliton microcomb (a) The optical spectrum of a single soliton state with 2.3 milliwatts operation power. (b) RF spectrum (resolution bandwidth = 100 kHz) of the entire soliton comb conﬁrms a low-noise state. (c) Measured frequency dispersion belonging to the soliton forming mode family (TE00) is plotted versus the relative mode number. The red curve is a ﬁt using D1/2π = 358.663 GHz and D2/2π = 8 MHz. Simulation of the soliton mode families is plotted (green curve), and the simulation fairly agrees with the measurement results. (d) Upper panel presents pump power transmission versus tuning across a resonance used for the soliton formation. Lower panel shows comb power trace in which the pump laser scans over the resonance from the short wavelength (blue detuned) to the long wavelength (red detuned). The shaded region (orange) depicts the spectral region where the single soliton exists.

(e.g., minimal avoided-mode-crossings). The power requirement for soliton operation is inversely proportional to the total Q factor of the mode familyS7. We demonstrate the generation of a dissipative soliton microcomb in a SiC microresonator. Figure S4a shows the spectrum measured for a single-soliton state, and the soliton spectral shape follows the square of a hyperbolic secant function. Small spurs in the spectrum correlate with the avoided-mode-crossings in the mode dispersion spectrum (Fig. S4c), and the RF spectrum of the single-soliton state conﬁrms that it is a low-noise state (Fig. S4b). While tuning the laser through the resonance mode, the pump power transmission as well as the comb power (Fig. S4d) show a step transition from modulation instability (MI) and a chaotic comb state to a stable soliton comb state. The high Q SiC resonator enables a low operation power of the soliton microcomb of 2.3 mW: Table I compares operation powers of various chip-scale soliton devices.

Soliton operation power Material Q0 (M) FSR (GHz) (OPO threshold) (mW) Reference Si N 260 5 20 Ref [S8] 3 4 ∼ Si3N4 8 194 1.3 (1.1) Ref [S9] Si3N4 15 99 6.2 (1.7) Ref [S10] SiO2/Si3N4 120 15 28 (5) Ref [S11] LiNbO3 2.4 199.7 5.2 Ref [S12] AlGaAs 1.5 450 1.77 (0.07) Ref [S13] SiC 5.6 350 2.3 (0.51) This work

TABLE I: Comparison of integrated soliton device performance 11

IV. SOLITON CRYSTALS

Soliton crystals, temporally-ordered ensembles of soliton pulses, have been observed in various optical resonator platforms, and their dynamics as well as defect-free generation have been actively explored. We demonstrate soliton crystal states with 2- and 7-FSR comb spacing, corresponding to phase-locked lattice of 2 and 7 identical solitons, respectively. We characterize the soliton crystals through the analysis of their optical spectra, RF beatnote, and second-order photon correlations.

A. 2-FSR soliton crystal

Figure S5 shows the OSA spectrum and RF beatnote noise of the soliton crystal state that is studied in Fig. 5 of the main text. In the main text, the RF beatnote of a single resonator mode is shown; In Fig. S5, we show the RF spectrum of the whole comb. Sweeping the laser from blue- to red-detuned, we observed a transition from a broad and noisy RF signal corresponding to the modulation instability (MI) state, to a low-noise state, coinciding with the beginning of a discrete step in the transmission trace across the cavity resonance.

FIG. S5: 2-FSR soliton crystal state (a) OSA spectrum of the soliton crystal state. Inset: pump power transmission versus laser tuning when the pump laser wavelength is scanned from blue- to red-detuned across the pump resonance. (b) RF spectra (resolution bandwidth = 100 kHz) of the soliton comb (black) and MI comb (red).

B. Photonic molecule analysis

The second-order correlation matrix for the 2-FSR soliton crystal state (presented in Fig. 5c of the main text) was computed via LLE simulation and input-output theory using the following parameters:

• D2/2π = 3.65 MHz, obtained from FEM simulation, neglecting higher-order terms. A single perturbation of 30 MHz was introduced at mode µ = 2 to induce the formation of the soliton crystal stateS14. − − • For the pump mode (µ = 0), the intrinsic and coupling Q factors of 2.37 and 6.55 million, respectively, were used, extracted from the measured cold-cavity transmission spectrum. • For the other modes, intrinsic and coupling Q of 2.77 and 7.47 million, respectively, were used, corresponding to the mean of the measured Q factors for the modes within the laser scanning range (µ = 3 to +14). − • Pump power of 6.6 mW in the waveguide, corresponding to the experimentally-measured value. The result of the LLE simulation is shown in Fig. S6b. The simulated soliton crystal spectrum for the detuning of 330 MHz is shown in Fig. S6c. In the input-ouput theory model, the laser detuning (within the range of existence of the soliton state in the LLE simulation) is the only free parameter. The corresponding second-order photon correlations and EN matrices are shown in Fig. S6d. Negligible entanglement is thus predicted in the resonator mode basis for this soliton crystal state. However, entanglement can be recovered by selectively over-coupling the below-threshold modes via a photonic molecule configuration, shown in Fig. S6e. This configuration is as follows: The auxillary resonator has a FSR that is 2 times larger than the FSR of the primary microring. The coupling strength of the two ring resonators exceeds the total losses (scattering and waveguide coupling) of the primary resonator. The auxillary ring is further over-coupled to its output waveguide, so that rather than be strongly-coupled to the primary resonator, the auxillary 12 resonator acts as a selective out-coupling channel for the odd-numbered modes of the primary resonator. We note that the finesse of the experimentally demonstrated resonators (approximately 3500) is sufficient for this architecture. To model this system, we perform the LLE simulation with the same device parameters as for the experimentally demonstrated device, but with the out-coupling rates of the odd modes increased by 10 times. We numerically confirm that the same 2-FSR soliton state can be prepared for this device (Fig. S6f). The computed second-order correlation and EN matrices for the the quantum state of this device are shown in Fig. S6h.

FIG. S6: Photonic molecule architecture for all-to-all entanglement generation (a) Schematic of the experimentally-demonstrated device. (b) LLE simulation of the device for pump power of 6.6 mW in the waveguide. The desired 2-FSR soliton crystal state exists for detuning in the range 175–395 MHz. This simulated soliton step width of 220 MHz is somewhat larger than the experimentally-observed step width of 150 MHz. (c) The simulated spectrum taken at detuning of 330 MHz. (d) Left: The two-photon correlation matrix computed for the state (2) in (c). The scale bar indicates max g (τ) . Right: The corresponding entanglement negativity, EN , matrix. (e-h) correspond to (a-d) but for the photonic{ molecule} configuration, where the out-coupling of the odd resonator modes is increased by 10 times via the auxilliary resonator (e) Schematic of the photonic molecule configuration. (f) It is confirmed via LLE that the same 2-FSR soliton crystal state can be captured in simulation. (g) The spectrum of the comb at the same detuning of 330 MHz is identical to the comb spectrum in the unmodified device, since only the below-threshold modes are affected by the addition of the auxiliary ring. (h) The corresponding correlation and entanglement matrices for the photonic molecule device. 13

C. 7-FSR soliton crystal

Figure S7 presents the generation of a 7-FSR soliton crystal state in a diﬀerent device. The optical spectrum (Fig. S7a) as well as the transmission and comb power traces across the pump resonance identify the existence of the soliton state. The SPOSA spectrum (Fig. S7c) reveals quantum frequency comb lines which were obscured by the noise ﬂoor of the OSA spectrum, and their correlation matrix is presented with the prediction from the LLE-driven linearized model (Fig. S7d).

FIG. S7: 7-FSR soliton crystal state (a) OSA spectrum of the soliton crystal state. (b) Pump power transmission (upper panel) and comb power (lower panel) versus wavelength tuning when the pump laser is scanned from blue to red across the pump resonance. (c) Optical spectrum of the soliton state measured using the SPOSA. (d) The max[g(2)(τ)] correlation matrix for the below threshold modes in the 7-FSR soliton state (Left: theoretical model, Right: experimental data).

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